Global existence of solutions to a hyperbolic-parabolic system

By Mei Zhang, Changjiang Zhu. 2007

94 citations recorded by the source corpus.

In this paper, we investigate the global existence of solutions to a hyperbolic-parabolic model of chemotaxis arising in the theory of reinforced random walks. To get L 2 -estimates of solutions, we construct a nonnegative convex entropy of the corresponding hyperbolic system. The higher energy estimates are obtained by the energy method and a priori assumptions.

Subfields: Cell Biology, Modeling and Simulation, Molecular Biology

View Paper: https://www.semanticscholar.org/paper/da734b6e8d0822983324c0fc2371933d929b58c7

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Citations in the corpus

Mathematics of traveling waves in chemotaxis --Review paper-- (2012, background): “2) on other aspects, such as self-similar solutions and collapse behavior [67], the global existence and large time behavior of solutions [90, 26, 46, 84, 79], and Cauchy problem [45].”
Well-posedness of a 3D parabolic–hyperbolic Keller–Segel system in the Sobolev space framework (2012, unknown): “[5, 7, 8, 10, 13, 14, 15, 16, 21, 22, 23, 25, 26, 27, 28, 30, 31, 34, 35, 36, 37] for more discussions in this direction.”
The two-dimensional Keller–Segel system with singular sensitivity and signal absorption: Global large-data solutions and their relaxation properties (2016, background): “1) obtained on replacing the second equation therein by the ODE vt = −uv, global existence results seem restricted to one-dimensional cases so far, whereas corresponding statements in higher-dimensional situations have been derived only under suitable smallness conditions on the data ([48], [25]).”
Asymptotic stability of traveling waves of a chemotaxis model with singular sensitivity (2013, unknown): “3) in one dimension were studied in [26] and in [1], respectively.”
Initial–boundary value problems for a system of hyperbolic balance laws arising from chemotaxis (2015, background): “[6,8–10,17,18,29,30,32] for blowup criteria, global existence and long-time behavior of classical solutions, and [19–21] for existence and local nonlinear stability of one-dimensional traveling wave solutions.”
LARGE-TIME BEHAVIOR OF A PARABOLIC-PARABOLIC CHEMOTAXIS MODEL WITH LOGARITHMIC SENSITIVITY IN ONE DIMENSION (2012, unknown): “3) in one dimension was studied in [45] and in [12], respectively.”
Boundary spike‐layer solutions of the singular Keller–Segel system: existence and stability (2019, background): “[3, 4, 6, 13, 21, 23–26]), global well-posedness and/or asymptotic behavior of solutions (see [5, 8, 19, 20, 22, 28, 33, 37, 42, 43] in one dimensional bounded or unbounded space and [7, 9, 18, 22, 32, 35, 39, 41] in multidimensional spaces) and boundary layer solutions [10–12].”